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SOA Fundamentals of Actuarial Mathematics Examination Practice Test

The SOA Fundamentals of Actuarial Mathematics Examination (FAM) practice page covers financial mathematics and life contingencies. Its question bank includes interest-rate conversions, annuity and loan values, bond yields, and the valuation of life insurance and annuity benefits using survival or death probabilities.

Exam overview

Exam providerSociety of Actuaries
Exam codeFAM
CategoryActuarial Examinations

About the SOA Fundamentals of Actuarial Mathematics Examination Exam

Questions call for translating contract terms and timing into cash-flow models, selecting discount or accumulation factors, and calculating present or accumulated values. Financial mathematics problems include level payments, outstanding loan balances, perpetuities, and effective versus nominal rates. Life-contingency problems add uncertainty about future lifetime: you may calculate expected present values for death benefits, represent a life annuity using curtate future lifetime, or find the variance of a benefit whose value depends on time of death. These tasks connect probability with discounting, requiring careful attention to when payments occur, which event triggers a benefit, and whether a value is certain or random. The resulting methods support actuarial work in pricing and evaluating insurance and annuity cash flows, as well as checking how assumptions about interest and mortality affect a valuation.

Sample Practice Questions

Review a selection from this exam's published US Practice Tests practice bank.

Question 1

A perpetuity-immediate pays 1 at the end of each year. At an effective annual interest rate of 4%, what is its present value?

  1. A24
  2. B25
  3. C26
  4. D27
Show answerCorrect answer: A

The present value of a perpetuity-immediate is 1/i. With i=0.04, the value is 1/0.04=25. The correct option is B.

Question 2

A 1,000 bond with 5 years remaining has annual coupons of 40 and is priced at par. Assuming annual coupons and redemption at face value, what is its annual effective yield?

  1. A3%
  2. B4%
  3. C5%
  4. D6%
Show answerCorrect answer: B

At a 4% yield, the present value of the 40 annual coupons plus the 1,000 redemption is 40a-angle-5 at 4% + 1,000(1.04)^−5=1,000. A coupon bond priced at par has yield equal to its coupon rate.

Question 3

An investment of 1,000 earns 5% effective interest per year. What is its accumulated value after 3 years?

  1. A1,150.00
  2. B1,157.63
  3. C1,165.00
  4. D1,200.00
Show answerCorrect answer: B

With annual effective interest, the accumulated value is 1,000(1.05)^3 = 1,157.625, or 1,157.63 rounded to cents.

Question 4

A nominal annual interest rate of 9% is convertible quarterly. What is the equivalent nominal annual interest rate convertible monthly?

  1. A8.89%
  2. B9.06%
  3. C9.14%
  4. D9.27%
Show answerCorrect answer: B

The quarterly rate is 0.09/4=0.0225, so the annual accumulation is (1.0225)^4. The equivalent monthly nominal rate j^(12) satisfies (1+j^(12)/12)^12=(1.0225)^4, giving j^(12)=12[(1.0225)^(1/3)-1]≈0.0906, or 9.06%.

Question 5

A benefit of 10,000 is payable at the moment of death, with present value Z=10,000e^(−0.05T). If E[e^(−0.05T)]=0.6 and E[e^(−0.10T)]=0.4, what is Var(Z)?

  1. A4,000,000
  2. B40,000,000
  3. C100,000,000
  4. D160,000,000
Show answerCorrect answer: B

E[Z]=10,000(0.6)=6,000 and E[Z²]=100,000,000(0.4)=40,000,000. Therefore Var(Z)=E[Z²]−(E[Z])²=40,000,000−36,000,000=4,000,000. The correct value is 4,000,000, option A.

Question 6

A life insurance benefit of 20,000 is payable at the end of the year of death if death occurs within 3 years; otherwise no benefit is paid. The effective annual interest rate is 5%, and the probabilities of death in years 1, 2, and 3 are 0.01, 0.02, and 0.03, respectively. What is the expected present value of the benefit?

  1. A1,063.78
  2. B1,200.00
  3. C1,141.76
  4. D1,063.78 plus a premium
Show answerCorrect answer: A

Discount each year-end benefit by its payment time and weight by the probability of death in that year: 20,000[0.01/1.05 + 0.02/(1.05)² + 0.03/(1.05)³] ≈ 1,063.78.

Question 7

A loan of 10,000 is repaid by equal annual payments at the end of each year for 4 years. The effective annual interest rate is 5%. Immediately after the first payment, what is the outstanding balance using the prospective method?

  1. A7,500.00
  2. B7,809.76
  3. C8,750.00
  4. D10,000.00
Show answerCorrect answer: B

The annual payment is 10,000/a-angle-4 at 5%, approximately 2,820.12. After the first payment, three payments remain, so the balance is 2,820.12 × a-angle-3 at 5% ≈ 7,809.76.

How to use this practice test

Work through free FAM practice questions in sessions of up to 20 randomly selected items from the available bank. After each session, review the displayed results and revisit the calculations that caused difficulty, such as payment timing, rate conversion, or discounting benefits contingent on survival or death.

Preparation tips

  • For each cash-flow problem, draw a timeline and mark payment dates, benefit triggers, and the valuation date before calculating.
  • Practice converting nominal rates by matching the accumulation over a common period; distinguish nominal rates from effective annual rates.
  • For loan questions, calculate the level payment first, then use the prospective value of remaining payments to find the balance after a payment.
  • For life-contingency calculations, write the probability associated with each payment time and discount each possible benefit to the valuation date.
  • For random present values, calculate the first and second moments separately; use Var(Z) = E[Z²] − (E[Z])².
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